1. The problem asks to construct a square MNOP with a diagonal length of 6 cm and then measure and write down the length of segment MN. 2. Important property: In a square, all sides are equal, and the diagonal length $d$ relates to the side length $s$ by the formula $$d = s\sqrt{2}$$ because the diagonal forms a right triangle with two sides. 3. Given the diagonal length $d = 6$ cm, we can find the side length $s$ by rearranging the formula: $$s = \frac{d}{\sqrt{2}}$$ 4. Substitute $d = 6$: $$s = \frac{6}{\sqrt{2}}$$ 5. To simplify, multiply numerator and denominator by $\sqrt{2}$: $$s = \frac{6}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{6\sqrt{2}}{\cancel{\sqrt{2}}\cancel{\sqrt{2}}} = 3\sqrt{2}$$ 6. Numerically, $3\sqrt{2} \approx 3 \times 1.414 = 4.242$ cm. 7. Therefore, the length of side MN is approximately 4.24 cm. 8. To construct the square: - Draw a diagonal segment of length 6 cm. - Use a compass to draw arcs from each endpoint with radius $3\sqrt{2}$ cm. - The intersection points of these arcs will be the other two vertices. - Connect all vertices to form square MNOP. Final answer: The length of side MN is $3\sqrt{2}$ cm, approximately 4.24 cm.